Step 1: Understanding the Concept:
To evaluate composite trigonometric functions, always start from the innermost bracket and move outwards.
We use principal values for inverse trigonometric functions.
For $\sin^{-1} x$, the principal branch is $[-\frac{\pi}{2}, \frac{\pi}{2}]$.
For $cot^{-1} x$, the principal branch is $(0, \pi)$.
Step 2: Key Formula or Approach:
1. Evaluate the inner $\sin^{-1}(1/2)$.
2. Multiply the result by 2.
3. Compute the cosine of the resulting angle.
4. Multiply by the outer coefficient 2.
5. Finally, find the inverse cotangent.
Step 3: Detailed Explanation:
Let the expression be $E = cot^{-1} [2 \cos (2 \sin^{-1} \frac{1}{2})]$.
Step i: Innermost part is $\sin^{-1} \frac{1}{2}$.
Since $\sin(\frac{\pi}{6}) = \frac{1}{2}$, we have $\sin^{-1} \frac{1}{2} = \frac{\pi}{6}$.
Step ii: Multiply by 2.
$2 \cdot (\frac{\pi}{6}) = \frac{\pi}{3}$.
Step iii: Compute the cosine of this angle.
$\cos(\frac{\pi}{3}) = \frac{1}{2}$.
Step iv: Multiply by the outer scalar 2.
$2 \cdot (\frac{1}{2}) = 1$.
Step v: Evaluate the outermost function $cot^{-1}(1)$.
We look for an angle $\theta$ such that $cot(\theta) = 1$ in the interval $(0, \pi)$.
Since $cot(\frac{\pi}{4}) = 1$, the result is $\frac{\pi}{4}$.
Therefore, $E = \frac{\pi}{4}$.
Step 4: Final Answer:
The value of the expression is $\frac{\pi}{4}$.